Show $\sup_{x > a}\frac{\sin x}{x} < 1$

Question

Let $a > 0$. I'm trying to show that

$$\sup_{x > a}\frac{\sin x}{x} < 1$$

Of course, showing it is $\leq$ is not hard: one uses that $\sin x < x$ for $x > 0$. Looking at the graph, this is quite obvious, but I can't make it formal.

Answer 1

Suppose, by contradiction, that the sup is $1$. That means there is a sequence $x_n$ with $x_n\ge a>0$ such that $\sin x_n/x_n\to 1$ as $n\to\infty$. This sequence either contains a subsequence convergent to some finite $L>0$ or a subsequence diverging to infinity. The latter is impossible because if $x>2$, $\sin x/x<1/2$. In the former case, by continuity you would have $\sin L/L=1$ which is false.